多次元システムの状態空間モデルの低次数化

秋田県立大学令和元年

Geometric Fault Detection and Isolation of Infinite Dimensional Systems

A broad class of dynamical systems from chemical processes to flexible mechanical structures, heat transfer and compression processes in gas turbine engines are represented by a set of partial differential equations (PDE). These systems are known as infinite dimensional (Inf-D) systems. Most of Inf-D systems, including PDEs and time-delayed systems can be represented by a differential equation in an appropriate Hilbert space. These Hilbert spaces are essentially Inf-D vector spaces, and therefore, they are utilized to represent Inf-D dynamical systems. Inf-D systems have been investigated by invoking two schemes, namely approximate and exact methods. Both approaches extend the control theory of ordinary differential equation (ODE) systems to Inf-D systems, however by utilizing two different methodologies. In the former approach, one needs to first approximate the original Inf-D system by an ODE system (e.g. by using finite element or finite difference methods) and then apply the established control theory of ODEs to the approximated model. On the other hand, in the exact approach, one investigates the Inf-D system without using any approximation. In other words, one first represents the system as an Inf-D system and then investigates it in the corresponding Inf-D Hilbert space by extending and generalizing the available results of finite-dimensional (Fin-D) control theory. It is well-known that one of the challenging issues in control theory is development of algorithms such that the controlled system can maintain the required performance even in presence of faults. In the literature, this property is known as fault tolerant control. The fault detection and isolation (FDI) analysis is the first step in order to achieve this goal. For Inf-D systems, the currently available results on the FDI problem are quite limited and restricted. This thesis is mainly concerned with the FDI problem of the linear Inf-D systems by using both approximate and exact approaches based on the geometric control theory of Fin-D and Inf-D systems. This thesis addresses this problem by developing a geometric FDI framework for Inf-D systems. Moreover, we implement and demonstrate a methodology for applying our results to mathematical models of a heat transfer and a two-component reaction-diffusion processes. In this thesis, we first investigate the development of an FDI scheme for discrete-time multi-dimensional (nD) systems that represent approximate models for Inf-D systems. The basic invariant subspaces including unobservable and unobservability subspaces of one-dimensional (1D) systems are extended to nD models. Sufficient conditions for solvability of the FDI problem are provided, where an LMI-based approach is also derived for the observer design. The capability of our proposed FDI methodology is demonstrated through numerical simulation results to an approximation of a hyperbolic partial differential equation system of a heat exchanger that is represented as a two-dimensional (2D) system. In the second part, an FDI methodology for the Riesz spectral (RS) system is investigated. RS systems represent a large class of parabolic and hyperbolic PDE in Inf-D systems framework. This part is mainly concerned with the equivalence of different types of invariant subspaces as defined for RS systems. Necessary and sufficient conditions for solvability of the FDI problem are developed. Moreover, for a subclass of RS systems, we first provide algorithms (for computing the invariant subspaces) that converge in a finite and known number of steps and then derive the necessary and sufficient conditions for solvability of the FDI problem. Finally, by generalizing the results that are developed for RS systems necessary and sufficient conditions for solvability of the FDI problem in a general Inf-D system are derived. Particularly, we first address invariant subspaces of Fin-D systems from a new point of view by invoking resolvent operators. This approach enables one to extend the previous Fin-D results to Inf-D systems. Particularly, necessary and sufficient conditions for equivalence of various types of conditioned and controlled invariant subspaces of Inf-D systems are obtained. Duality properties of Inf-D systems are then investigated. By introducing unobservability subspaces for Inf-D systems the FDI problem is formally formulated, and necessary and sufficient conditions for solvability of the FDI problem are provided

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Measurements, Models, Systems and Design

531 s.

Iterative Learning Control design for uncertain and time-windowed systems

Iterative Learning Control (ILC) is a control strategy capable of dramatically increasing the performance of systems that perform batch repetitive tasks. This performance improvement is achieved by iteratively updating the command signal, using measured error data from previous trials, i.e., by learning from past experience. This thesis deals with ILC for time-windowed and uncertain systems. With the term "time-windowed systems", we mean systems in which actuation and measurement time intervals differ. With "uncertain systems", we refer to systems whose behavior is represented by incomplete or inaccurate models. To study the ILC design issues for time-windowed systems, we consider the task of residual vibration suppression in point-to-point motion problems. In this application, time windows are used to modify the original system to comply with the task. With the properties of the time-windowed system resulting in nonconverging behavior of the original ILC controlled system, we introduce a novel ILC design framework in which convergence can be achieved. Additionally, this framework reveals new design freedom in ILC for point-to-point motion problems, which is unknown in "standard" ILC. Theoretical results concerning the problem formulation and control design for these systems are supported by experimental results on a SISO and MIMO flexible structure. The analysis and design results of ILC for time-windowed systems are subsequently extended to the whole class of linear systems whose input and output are filtered with basis functions (which include time windows). Analysis and design theory of ILC for this class of systems reveals how different ILC objectives can be reached by design of separate parts of the ILC controller. Our research on ILC for uncertain systems is divided into two parts. In the first part, we formulate an approach to analyze the robustness properties of existing ILC controllers, using well developed µ theory. To exemplify our findings, we analyze the robustness properties of linear quadratic (LQ) norm optimal ILC controllers. Moreover, we show that the approach is applicable to the class of linear trial invariant ILC controlled systems with basis functions. In the second part, we present a finite time interval robust ILC control strategy that is robust against model uncertainty as given by an additive uncertainty model. For that, we exploit H1 control theory, however, modified such that the controller is not restricted to be causal and operates on a finite time interval. Furthermore, we optimize the robust controller so as to optimize performance while remaining robustly monotonically convergent. By means of experiments on a SISO flexible system, we show that this control strategy can indeed outperform LQ norm optimal ILC and causal robust ILC control strategies

MODELING OF MHD INSTABILITIES IN EXISTING AND FUTURE FUSION DEVICES IN VIEW OF CONTROL

In questo lavoro viene presentata una versione migliorata del codice CarMa, chiamato CarMa-D, per lo studio di Resistive Wall Modes (RWMs) nei reattori a fusione termonucleare. Tale codice \ue8 in grado di rappresentare accuratamente le strutture conduttrici tridimensionali della macchina, e considerare simultaneamente nel modello gli effetti dovuti alla dinamica del plasma, alla toroidal rotation e agli effetti drift-cinetici. CarMa-D \ue8 il risultato dell\u2019accoppiamento dei codici CARIDDI, per lo studio delle correnti indotte nelle strutture conduttrici, e MARS-K per analisi di stabilit\ue0 MHD nel plasma. Punto di forza della strategia di accoppiamento alla base di CarMa-D \ue8 che non si basa sulle ipotesi semplificative su cui si basa la versione statica di CarMa, ovvero non vengono trascurati la massa del plasma, toroidal rotation e l\u2019effetto del damping cinetico. In questo modo la risposta del plasma a perturbazioni esterne dipende dall\u2019andamento temporale della perturbazione stessa: questo andamento viene approssimato per mezzo di funzioni razionali di Pad\ue9 a coefficienti matriciali. Il passo successivo \ue8 dato dalla combinazione della risposta di plasma approssimata con l\u2019equazione delle correnti indotte nelle strutture passive, per ottenere un modello matematico desctitto come un sistema di equazioni differenziali lineari formalmente uguale alla versione statica di CarMa, ma con un numero maggiori di gradi di libert\ue0 per tener conto della dinamica di plasma. La nuova versione del codice supera le principali limitazioni del modello originale, in particolare: (i) considerando la massa del plasma \ue8 possibile modellare modi con dinamiche molto veloci, come l\u2019external-kink ideale, (ii) il modello \ue8 in grado di tener conto rigorosamente di toroidal rotation e damping cinetico. Questi vantaggi rendono CarMa-D uno strumento potente, in grado di studiare fenomeni macroscopici in cui sia la dinamica del plasma, che gli effetti 3-D delle strutture, sono marcati. Inoltre, il modello matematico risultate \ue8 stato generalizzato per tener conto della simulazione pi\uf9 armoniche toroidali simultaneamente (multi-modal CarMa-D). Il codice \ue8 stato poi testato con successo su un equilibrio di riferimento dato da un plasma a sezione circolare, e successivamente per lo studio di stabilit\ue0 per i modi n = 1 e n = 2 su JT-60SA, Scenario 5. Infine, si \ue8 dimostrato come il modello matematico di CarMa-D possa essere scritto in una formulazione state-space, in vista di un successivo impiego nella progettazione di un controllo in retoazione per la stabilizzazione attiva dei RWMs